This paper introduces an extension of minimum variance beamforming, also known
as Capon's method, that explicitly takes into account variation or uncertainty
in the assumed array response. Sources of this uncertainty include imprecise
knowledge of the angle of arrival and uncertainty in the array manifold. In
Capon's method, the weights are chosen to minimize the weighted array power
output subject to a unity gain constraint in the desired look direction. This
method assumes that the array manifold is precisely known; unfortunately, even
small variations in the array manifold can drastically reduce its performance.
In our method, uncertainty in the array manifold is explicitly modeled via an
emph{uncertainty ellipsoid} that gives the possible values of the array for a
particular look direction. We choose weights that minimize the total weighted
power output of the array, subject to the constraint that the gain should
exceed unity for all array responses in this uncertainty ellipsoid. If the
ellipsoid reduces to a single point, the method coincides with Capon's method.
Unlike Capon's method, however, we can guarantee performance of the robust
method in the presence of uncertainties. We show that the robust weights can be
computed efficiently using Lagrange multiplier techniques. In fact, the robust
weight selection problem can be solved in the same order of complexity as the
non-robust counterpart. We describe in detail several methods that can be used
to derive an appropriate uncertainty ellipsoid for the array response. The
simplest methods fit an ellipsoid around empirical (or simulated) data. In a
more sophisticated approach, we form separate uncertainty ellipsoids for each
component in the signal path (e.g., antenna, electronics) and then determine an
aggregate uncertainty ellipsoid from these. We give new results for modeling
the element-wise products of ellipsoids. We demonstrate the robust beamforming
and the ellipsoidal modeling methods with several numerical examples.